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REVIEW 4 major objections 6 minor 23 references

Dynamic Programming-Based Redundancy Resolution for Path Planning of Redundant Manipulators Considering Breakpoints

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Dynamic programming finds the globally best joint angles for a redundant arm, interruptions included.

desk verdict Useful DP extension for redundancy resolution, but the feasibility guarantee is not enforced for rest-to-motion transitions and the formal loss function is backwards. read the letter →

arxiv 2411.17034 v1 pith:M66R6CVP submitted 2024-11-26 cs.RO

Zhihang Yin , Fa Wu , Ruofan Bian , Ziqian Wang , Jianmin Yang , Jiyong Tan , Dexing Kong This is my paper · ORCID
classification cs.RO
keywords redundancyresolutiondynamicprogrammingredundantmanipulatorinversekinematicspathplanningbreakpointminimizationoptimalcontroltrajectoryoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that redundancy resolution along a pre-planned Cartesian path becomes a tractable global optimization when the redundant joint is treated as a discrete parameter and the search is handed to dynamic programming. Fixing joint 7 as the parameter and using an analytical inverse-kinematics map, each sampled pose on the path yields a finite set of candidate joint configurations; the DP then scans all sequences of these candidates under hard joint-angle, velocity, and acceleration bounds and returns the sequence with the lowest loss. The loss function adds a large penalty to every interruption, so minimizing it first minimizes the number of breakpoints and only then the sum of squared joint displacements, which means a 7-DOF arm can follow the path with the fewest stops and the least joint motion among all discretized choices of the redundancy parameter. If no fully continuous joint-space path exists, the same DP outputs the minimum number of breakpoints and their optimal locations, and a variant for closed circular paths can shift the starting point to reduce interruptions, demonstrated in the paper on a test path where the breakpoint disappears entirely. A curious reader should care because the alternative, a local per-pose solver, stalls the real Franka arm mid-scan in the paper's experiments, whereas the DP method completes the motion.

What carries the argument

The central object is the discretized, parameterized inverse-kinematics map $\tilde f^{-1}(\mathbf{T}_{EE}, q_7)$, which fixes the seventh joint angle as the redundancy parameter and returns the remaining six angles in closed form, with the branch-elimination rule of the cited solver making the map bijective over feasible poses. The paper spans the $q_7$ range with $m$ discrete values $\{a_1,\dots,a_m\}$, producing an $m \times (n+1)$ grid of candidate joint configurations $\bar{\mathbf{q}}_{i,j}$ for the $n$ sampling instants. The dynamic program then works on this grid: $\tilde L(i,j,k)$ is the minimum loss of a partial path ending at configuration $(i,j)$ whose previous configuration is $(i-1,k)$; transitions are assigned infinite cost when the implied velocity or acceleration exceeds the joint limits, and an interruption is represented by the penalty term $M$ in equations (16)-(18), which restarts the recurrence from the best configuration two steps back. The loss function $L(\{\mathbf{q}_a\}) = \sum_{i=1}^{a} \left[\operatorname{cont}_i \|\mathbf{q}_i - \mathbf{q}_{i-1}\|_2^2 + (1 - \operatorname{cont}_i)M\right]$, with $M > n\|\mathbf{q}_{\max} - \mathbf{q}_{\min}\|_2^2$, is what carries the argument: $L/M$ equals the number of interruptions, so minimizing $L$ settles the breakpoint count first and the summed squared joint displacement second, and backtracking through the minimizing $p$ in $\hat L$ recovers the optimal joint sequence.

What would settle it

Take any pose on the test path and any $q_7$ value inside the claimed feasible band, and independently enumerate all joint configurations reaching that pose with a different method, such as a numerical IK solver with many random restarts or a certified global inverse-kinematics solver; if any configuration appears that the parameterized map $\tilde f^{-1}(\mathbf{T}_{EE}, q_7)$ does not produce, or any configuration it produces violates a joint limit in Table 1, the state-space completeness premise fails and the DP may declare a traversable path infeasible. A second, cheaper check on the same premise: rerun the DP on the test paths with $m$ substantially larger than 4000; if the number of breakpoints ever decreases, the reported minimum was an artifact of the discretization rather than a property of the path.

Watch

Extended reading notes

Core claim

In the paper's own terms, the discovery is that redundancy resolution for a 7-DOF manipulator on a discretized Cartesian path has a globally optimal solution that can be computed by a dynamic program over an $m \times (n+1)$ grid of inverse-kinematics configurations. The parameterization $\tilde f^{-1}(\mathbf{T}_{EE}, q_7)$ from the cited analytical solver makes every pose correspond to a curve of solutions indexed by joint 7; discretizing $q_7$ into $m$ values turns the continuous redundancy into a finite choice at each of the $n$ sampling instants. The recurrence $\tilde L(i,j,k) = \min_p \hat L(i,j,k,p)$ propagates the minimal loss through triples of consecutive configurations, rejecting transitions that violate the velocity or acceleration limits and charging the large constant $M$ at every interruption; the output $\min_{j,k} \tilde L(n,j,k)$ is the globally minimal loss over this grid. Because $M > n\|\mathbf{q}_{\max} - \mathbf{q}_{\min}\|^2$, dividing the loss by $M$ counts the interruptions, so the same algorithm that minimizes joint motion also provably returns the minimum number of breakpoints when a continuous traversal is impossible, and Algorithm 2's modified cost on the doubled path identifies a new starting point that cuts one interruption from a closed circular path whenever such a cut exists.

Load-bearing premise

The load-bearing premise is that fixing joint 7 as the redundancy parameter and applying the cited analytical solver's branch-elimination rule yields every feasible joint configuration for every pose on the path; if that map misses some valid configurations or admits configurations that violate a joint limit, the DP's 'global' optimum is only optimal over an incomplete subset of the joint space.

Editorial extensions

If this is right

  • A 7-DOF arm can follow a presampled Cartesian path with the fewest interruptions and the smallest sum of squared joint displacements among all discretized choices of the redundancy parameter, so a scanning or polishing task runs with the minimum number of stops.
  • When the path cannot be traversed continuously, the same recurrence returns the minimum number of breakpoints needed and the best places to interrupt, instead of halting at the first pose where the local solver fails.
  • For closed circular paths, the starting-point modification provably reduces the required number of interruptions by at most one, and it gives a concrete criterion for when that reduction is achievable; in the paper's test it eliminates the interruption entirely.
  • Computing the DP on a sparse grid (10 points per second) and filling in the rest with the interpolation and motion-compensation layer keeps all angle, velocity, acceleration, and jerk constraints satisfied, with average Cartesian errors near $10^{-6}$ m.
  • The scheme transfers to manipulators with more than 7 degrees of freedom by replacing the scalar parameter $q_7$ with a parameter vector, enlarging the grid correspondingly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The penalty construction makes the algorithm a minimum-cardinality segmentation routine in disguise: minimizing $L/M$ first picks the smallest number of feasible continuous segments covering the path, so the same DP skeleton would solve other segmentation objectives (energy use, patient contact force during an ultrasound sweep) by swapping the per-segment cost term.
  • The paper's 'globally optimal' statement is relative to the fixed discretization $m$; the authors note that larger $m$ lowers the loss, which implies the true continuous optimum is approached rather than attained, so a post-processing continuous refinement of $q_7$ between DP-selected grid values could yield grid-independent optima.
  • For ultrasound or massage robots that work on body-surface paths, the starting-point result gives an operational principle: choose the probe's entry point to coincide with the first interruption of the unmodified circular scan, and the whole examination can run without re-orienting the arm.
  • Because the breakpoint count is read off the topological structure of the existence regions in the $(t, q_7)$ plane, the DP could in principle be replaced by a reachability analysis on that graph for the breakpoint count alone, reserving the DP for the fine-grained motion cost.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a dynamic-programming (DP) redundancy-resolution algorithm for the 7-DOF Franka Emika manipulator. The Cartesian path is sampled at fixed intervals; at each pose, candidate joint configurations are generated by discretizing the redundancy parameter q7 into m values using the analytic inverse-kinematics solver of [24] (Eqs. (6)-(7)). A loss function (Eq. (5)) combines squared joint displacement for continuous transitions with a large penalty M per breakpoint, so that minimizing it lexicographically minimizes first the number of breakpoints and then joint motion. A DP over the configuration grid (Eqs. (10)-(19), Algorithm 1) is claimed to yield the globally optimal joint-space path subject to angle, velocity, and acceleration constraints; when no fully continuous path exists, the penalty structure yields the minimum number and optimal placement of breakpoints. Algorithm 2 shifts the starting point of circular paths to reduce breakpoints. Since the DP runs at 100 path points per second while the controller operates at 1 kHz, Section 3.4 adds linear interpolation with a real-time compensation loop (Algorithm 3) that clamps jerk, acceleration, and velocity. Section 4 reports simulated and hardware experiments comparing the DP with Franka's Cartesian pose generator, an interpolation error analysis, and a demonstration of starting-point modification.

Significance. The paper addresses a practically motivated problem (ultrasound scanning) and its formulations have real merit. The penalty-based loss (Eq. (5)) with M > n||q_max - q_min||^2 correctly forces the DP (Eqs. (15)-(17)) to minimize the number of breakpoints before the displacement cost; the breakpoint-placement optimization and the starting-point modification for circular paths (Algorithm 2, Section 3.3) are useful ideas beyond the existing DP-based redundancy-resolution literature [5, 19]; and the hardware comparison against the local Franka solver (Section 4.1, Figure 6) is a convincing qualitative demonstration that local methods can fail on paths that a global planner can traverse. If the algorithm were fixed and re-validated, the paper would be a valuable contribution to path-level redundancy resolution. However, the central guarantee is not established as written: the DP omits the acceleration constraint for the first transition and for transitions after breakpoints (contradicting Eqs.

major comments (4)
  1. [Section 3.2, Eqs. (12) and (16)] The DP never enforces the acceleration constraint for transitions that start from rest. For i = 1, Eq. (12) assigns Ltilde(1,j,k) = ||qbar(1,j) - qbar(0,k)||^2 after only the velocity check of Eq. (11); the acceleration (qbar(1,j,c) - qbar(0,k,c))/t0^2 is left unbounded even though the robot is initially at rest. In the breakpoint branch, Eq. (16) interrupts the motion between qbar(i-2,p) and qbar(i-1,k) and then adds ||qbar(i,j) - qbar(i-1,k)||^2 without checking that, restarting from rest at qbar(i-1,k), the joints can reach qbar(i,j) in one sampling interval. Under the paper's own discrete definitions (Eqs. (3)-(4)), a transition from rest must satisfy |qbar(i,j,c) - qbar(i-1,k,c)| <= qddot_max,c * t0^2. With t0 = 0.01 s and qddot_max,2 = 7.5 rad/s^2, a joint-2 displacement of only 7.5e-4 rad already saturates the acceleration limit, whereas the velocity check alone permits displacements up to qdot_max,2 * t0 = 2.2e-2 rad; the DP can therefore certify transitions whose required acceleration is up to about 30 times the stated limit. This is not a modeling choice but an inconsistency with Eqs. (3)-(4), and it undermines the Section 4.1 claim that the algorithm guarantees the velocity and acceleration constraints and finds a globally optimal feasible path whenever one exists. The proof promised in 'appendix B' is not part of the manuscript, so I could not verify whether it covers the from-rest cases.
  2. [Section 3.1, Eq. (6)] The 'globally optimal' claim is contingent on the completeness of the q7-parameterized IK map f^{-1}_{q7} taken from [24]. The abstract asserts that the algorithm obtains 'all feasible inverse kinematic solutions for each pose under the joint angle constraints,' but the manuscript offers no verification that the branch-elimination rule of [24] covers every joint-limit-respecting configuration for every pose on the test paths, or that it never generates configurations violating the limits at specific poses. If the enumeration omits valid branches or includes invalid ones, the DP state space is incomplete or contaminated, and Figures 2-3, which motivate the breakpoint analysis, could misrepresent the feasible q7 regions. Since the DP optimizes only over {qbar(i,j)}, any appendix-B optimality claim is at best global over the enumerated discrete set. I recommend either an explicit completeness check against an independent complete IK solver (such as the polynomial method of [12]) on a dense pose grid, or a qualification of the optimality claim as relative to the discrete parameterization.
  3. [Section 3.4, Algorithm 3] The paper's guarantee statements concern the low-rate DP output, but the trajectory actually commanded to the robot is the output of the interpolation and real-time compensation loop. Section 3.4 concedes that linear interpolation alone violates the acceleration constraints, and Algorithm 3 enforces the limits by clamping jerk, acceleration, and velocity at each 1 kHz cycle, which changes the trajectory; Section 4.2 likewise concedes error accumulation at 100 path points per second (Figure 8). Hence the executed motion is not the DP-optimized path, and the claim in Section 4.2 that 'the constraints on the angles, velocities, accelerations and jerks of each joint have been duly satisfied' is a property of the simulated compensation loop, whose 'cautionary' velocity limit and reduced joint limits are introduced without derivation, rather than a consequence of Eqs. (10)-(19). The paper should either prove that the compensation tracks the planned knots within a bounded error while staying within all limits, or restrict the constraint guarantee to the DP-level plan and present the compensation as an empirically validated engineering approximation.
  4. [Algorithm 1, Section 3.2] The computational cost of Algorithm 1 is not stated, and the naive implementation is O(n*m^3): for each layer i and each pair (j,k) passing the velocity check, the inner loop over p computes Lhat(i,j,k,p). For a 10 s path at 100 path points per second (n = 1000) and m = 4000, as reported in Section 4.1, this is on the order of 10^13 operations, which is not plausible for the experiments as described. If the implementation exploits the structure of Eqs. (15)-(19), for example by precomputing min_p Ltilde(i-1,k,p) and min_r Ltilde(i-2,p,r) or by pruning with the velocity and acceleration filters, that reduction should be documented; otherwise the reported parameter setting is not reproducible. Please report the achieved complexity, wall-clock runtimes, and memory use per experiment.
minor comments (6)
  1. [Section 4.1 and Section 4.3] There are cross-reference errors in the experimental sections: Section 4.1 refers to 'the same Cartesian path TEE1(t) in equation (9)' although Eq. (9) defines TEE2 (TEE1 is defined in Eq. (8)), and Section 4.3 refers to 'the path TEE2(t) as defined in Equation (10)' although Eq. (10) defines the DP value Ltilde(i,j,k), not a path.
  2. [Eq. (5)] In Eq. (5) the loss term is printed as cont_i * ||q_i - q_{i-1}||^2 + (1 - cont_i) * M, which assigns the penalty M to continuous transitions and the squared displacement to interrupted transitions, the opposite of the semantics stated in the prose and implemented by the DP recurrences (Eqs. (15)-(17)). The roles of cont_i and (1 - cont_i) appear to be swapped; please correct the equation.
  3. [Section 2 and Section 3.4] The symbol t0 is overloaded: Section 2 defines t0 as the communication cycle (1 ms for the Franka), Section 4.1 uses 100 path points per second, implying a DP sampling interval of 10 ms, and Eq. (21) with Algorithm 3 again calls t0 the communication period. Please introduce distinct symbols, for example a DP sampling interval T_s and a controller period T_c, and state their relationship explicitly.
  4. [Eq. (22)] The stopping constraints in Eq. (22) appear dimensionally inconsistent: qddot_max^2 / (2 * qddot_max) has units of rad/s^2, not rad/s, so the second inequality cannot be a bound on qdot. Please re-derive or correct this formula, as a typo involving qddot_max and qdot_max seems likely.
  5. [Section 4.1, appendix reference] The global-optimality proof is referenced as 'appendix B' in Section 4.1, but no appendix is included in the manuscript; please include the proof or remove the reference, since the Section 4.1 guarantee depends on it.
  6. [Section 3.1] The bijectivity claim for f^{-1}_{q7} would be easier to evaluate if the workspace in which the map is bijective were stated precisely; as written, 'within the workspace of the 6-DOF manipulator' is ambiguous about the role of joint limits in defining that workspace.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DP optimization and breakpoint analysis are self-contained over an externally sourced IK parameterization.

full rationale

The paper's central derivation is a dynamic program over a finite set of inverse-kinematics configurations generated by a published analytical IK solver [24]. The loss function L({q_a}) in Eq. (5) is a defined objective, not a fitted quantity: the penalty M is chosen by the explicit inequality M > n||q_max - q_min||_2^2 so that minimizing L first minimizes the number of breakpoints and then the squared joint displacement. The DP recurrence (Eqs. 10-19) propagates these costs and prunes transitions using the joint velocity and acceleration inequalities from Eq. (4); no parameter is calibrated to data, and no prediction is statistically forced by a fitted input. The claimed global optimality is explicitly scoped to the discretized redundancy parameter q7 with m fixed values, as stated in Section 3.1 and Appendix B, so it is a genuine combinatorial optimum over that discrete state space rather than an equivalence-to-inputs. The starting-point modification argument in Section 3.3 is a direct counting proof about breakpoints, not a circular redefinition. The paper does rely on external prior work for the bijective IK map and for the DP technique, but these are independent baselines, not self-citations that smuggle in the target result. A reviewer concern that the algorithm does not check acceleration for the first transition or immediately after a breakpoint is a correctness or completeness issue, not circularity, because the omitted check is not an input that has been renamed as an output. Overall, no step in the derivation reduces by construction to its own assumptions.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests mainly on the completeness of the q7 parameterization and on the DP recurrence's treatment of breakpoints. There are no newly postulated physical entities. The free parameters are discretization and penalty choices, not fitted to data.

free parameters (3)
  • m, number of discrete q7 values = 4000 in experiments
    Chosen by hand; the claim of global optimality is only over this finite discretization. Larger m improves the approximation but increases computational cost; no convergence criterion is provided.
  • Penalty constant M = M > n ||qmax - qmin||^2
    Chosen to dominate all squared joint displacements; the inequality defines a lower bound but not a specific value. It is a modeling knob, not fitted to data.
  • Cautionary velocity limit and reduced joint limits in interpolation = Not specified
    Section 3.4 introduces a 'cautionary value' for velocity and says joint limits should be reduced to leave room for deceleration, but gives no rule for setting these margins. The behavior of the compensation loop depends on them.
assumptions (5)
  • standard math Dynamic programming optimality principle applies to the discretized redundancy resolution problem.
    The recurrence in Eqs. (15)-(19) relies on the loss having optimal substructure over the three-point transition window. This is standard, but the breakpoint branch changes the state definition, so the optimal substructure must be re-established; the paper does not prove it carefully.
  • domain assumption The analytic IK solver from [24] parameterized by q7 is complete and bijective after branch elimination.
    Section 3.1: the paper states that a bijective mapping f^{-1}_{q7} is obtained by excluding multiple solutions through joint constraints. This is assumed rather than derived; if any feasible configuration is missed, the DP state space is incomplete.
  • domain assumption Joint velocities and accelerations are adequately characterized by backward finite differences over the sampling interval t0.
    Equation (3) defines velocity and acceleration as first and second differences. This implicitly assumes piecewise-constant or piecewise-linear joint motion at the sample rate; the actual high-rate interpolated motion is not the same signal, and constraints on the true motion are not proven.
  • ad hoc to paper After a breakpoint, the manipulator can restart from the new joint configuration with zero velocity and follow the next sampled point in one communication cycle.
    The breakpoint branch in Eq. (16) charges penalty M and resumes motion at q_{i-1,k} without checking the acceleration needed to reach q_{i,j} from rest. This is an implicit physical feasibility assumption that is not validated and is likely violated for the stated limits.
  • ad hoc to paper Linear interpolation plus the real-time compensation loop preserves the feasibility of the low-rate DP solution.
    Section 3.4 and Algorithm 3 clamp jerk, acceleration, and velocity, but no proof is given that the resulting high-rate trajectory satisfies all constraints and tracks the Cartesian path; Section 4.2 acknowledges error accumulation.

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Cite this review

Pith. "Pith review of Dynamic Programming-Based Redundancy Resolution for Path Planning of Redundant Manipulators Considering Breakpoints." pith.science (2026). https://pith.science/paper/M66R6CVP

@misc{pith2026241117034,
  author       = {Pith},
  title        = {Pith review of: Dynamic Programming-Based Redundancy Resolution for Path Planning of Redundant Manipulators Considering Breakpoints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M66R6CVP}},
  note         = {Machine review of arXiv:2411.17034}
}
read the original abstract

This paper proposes a redundancy resolution algorithm for a redundant manipulator based on dynamic programming. This algorithm can compute the desired joint angles at each point on a pre-planned discrete path in Cartesian space, while ensuring that the angles, velocities, and accelerations of each joint do not exceed the manipulator's constraints. We obtain the analytical solution to the inverse kinematics problem of the manipulator using a parameterization method, transforming the redundancy resolution problem into an optimization problem of determining the parameters at each path point. The constraints on joint velocity and acceleration serve as constraints for the optimization problem. Then all feasible inverse kinematic solutions for each pose under the joint angle constraints of the manipulator are obtained through parameterization methods, and the globally optimal solution to this problem is obtained through the dynamic programming algorithm. On the other hand, if a feasible joint-space path satisfying the constraints does not exist, the proposed algorithm can compute the minimum number of breakpoints required for the path and partition the path with as few breakpoints as possible to facilitate the manipulator's operation along the path. The algorithm can also determine the optimal selection of breakpoints to minimize the global cost function, rather than simply interrupting when the manipulator is unable to continue operating. The proposed algorithm is tested using a manipulator produced by a certain manufacturer, demonstrating the effectiveness of the algorithm.

Figures

Figures reproduced from arXiv: 2411.17034 by the authors.

Figure 1
Figure 1. Denavit-Hartenberg frames and table of parameters for the Franka Emika robot. In this figure, joint angles 𝑞1,...,7 = 0. The reference frames follow the modified Denavit-Hartenberg convention, 𝑑1 = 0.333m, 𝑑3 = 0.316m, 𝑑5 = 0.384m, 𝑑𝑓 = 0.107m, 𝑎4 = 0.0825m, 𝑎5 = −0.0825m, 𝑎7 = 0.088m [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The existence of solutions corresponding to each 𝐓̂ EE1(𝑡) and 𝑞7 is depicted in a figure, where the blue regions indicate the existence of the corresponding inverse kinematic solutions 𝑓̃−1(𝐓̂ EE1(𝑡), 𝑞7 ). Taking into account the constraints on joint velocity and acceleration, the curve of 𝑞7 over time will be a continuous curve within the blue portion of the figure of the existence of inverse kinematic solutions.… view at source ↗
Figure 3
Figure 3. The existence of solutions corresponding to each 𝐓̂ EE2(𝑡) and 𝑞7 is depicted in a figure, where the blue regions indicate the existence of the corresponding inverse kinematic solutions 𝑓̃−1(𝐓̂ EE2(𝑡), 𝑞7 ). A continuous curve of 𝑞7 does not exist, therefore the manipulator cannot traverse from the start to the end along the path without interruption. From this example, it is evident that in designing the algorithm … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The computed path results of the proposed algorithm based on DP(dynamic programming) and the Franka Cartesian pose generator. The joint angle 𝑞7 obtained from the Franka Cartesian pose generator reaches its lower bound during the path, leading to the inability to conti…
Figure 5
Figure 5. Figure 5: The left plot depicts the manipulator path obtained from the proposed algorithm, closely aligning with the original path. On the right, the plot shows the motion trajectory of the manipulator under the Franka Cartesian pose generator, with its start and end points indi…
Figure 6
Figure 6. Figure 6: The top row illustrates the results obtained from the proposed algorithm, while the bottom row shows the outcomes of the Franka algorithm. Starting from the initial position, both algorithms select different poses, resulting in divergent final motion results. Under the…
Figure 7
Figure 7. Figure 7: The angles, velocities, accelerations and jerk of each joint are normalized to the range [-1, 1], where -1 and 1 respectively correspond to the minimum and maximum values of each parameter. The figure demonstrates that the constraints on the angles, velocities, acceler…
Figure 8
Figure 8. Figure 8: We conducted redundancy resolution for the test path using 100 path points per second and 10 path points per second respectively, and computed the error between the interpolated joint angles and the expected joint angles. At certain path points, the results obtained at…
Figure 9
Figure 9. Figure 9: The redundancy resolution of 𝐓̂ EE2(𝑡) computed with Algorithm 1. It is evident from the plot that the curves representing the joint variations over time exhibit a distinct discontinuity, indicating the presence of discontinuities in the motion [PITH_FULL_IMAGE:figure…
Figure 10
Figure 10. Figure 10: We employed Algorithm 2 to compute 𝐓̂ EE2(𝑡). The algorithm successfully identified a new starting point that reduced the number of discontinuities in the motion. Upon altering the starting point, the joint angles of the manipulator, as depicted in this figure, no lon…
Figure 11
Figure 11. Figure 11: The manipulator poses corresponding to the results obtained from both algorithms.The upper image corresponds to the outcome of Algorithm 1, revealing that the manipulator deviates significantly from the expected path midway, circling in a different direction before co…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.